Introduction to Group Theory, part 2
• vibrational modes of pyramidal XY3 (C3v) molecules
• further example: P(4) and the point group Td
• notation of point groups (Schoenflies and international)
• direct product of matrices and groups
• example of D3h = C3v C1h
• notation for the representations
• symmetry operations on functions of coordinates
• basis functions
Fyzika kond. látek 2, jarní semestr 2021/2022 J. Humlíček, humlicek@physics.muni.cz
1
Molecular vibrations - Herzberg II, pyramidal XY3 (C3v);
3N-6=6 vibrations allowed in both Raman and IR, two doubly degenerate
2
Example - Td
Symmetry operations of a regular tetrahedron (zincblende structures):
identity E,
eight C3 axes about diagonals (dashed lines),
three C2 axes about x,y,z,
six S4 axes about x,y,z, corresponding to
the rotations of ±p/2
six sd reflections (diagonal planes)
the group Td of the order of 24,
isomorphic with P(4),
5 classes, the character table is a 5x5 matrix
3
Example - Td
Character tables for the point group Td from two sources:
Inui, Tanabe, Onodera, Group theory and its applications in physics, Springer 1976
(43m)
M.S. Dresselhaus, G. Dresselhaus, A. Jorio, Group theory, Applications to the physics of Condensed Matter,
Springer 2008
4
Example - Td
Character tables for the point group Td and (isomorphic) permutation group P(4)
from Dress_2008. Try to understand the notation used for the elements of classes
of P(4), and their link to the symmetry operations of Td .
Note the different labels of the irreducible representations.
5
(Crystallographic) point groups – two main naming conventions:
Schoenflies and “international” (Hermann-Maguin)
Translation symmetry restricts the n-fold rotation axis Cn to n=1,2,3,4, and 6.
Schoenfliess international
Cn 1,2,3,4,6
s (mirror reflection) m
Sn (rotatory inversion axis)
the symbol m for the mirror plane does not distinguish between vertical,
horizontal, and diagonal planes; instead,
n/m means a horizontal plane perpendicular to the n-fold axis,
nm means a horizontal plane containing the n-fold axis.
1,3,4,6
6
Point groups – two main naming conventions for 32 crystallographic point groups
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Point groups – two main naming conventions for 32 crystallographic point groups
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Direct product of matrices
Let A and B be a matrices of lAc lAr and lBc lBr elements:
Aij , i=1,...,lAr , j=1,...,lAc , and Bkm , k=1,...,lBr, m=1,...,lBc .
The matrix C=AB, called the direct product, consists of lAr lAc lBr lBc elements of
all products Aij Bkm = Cik,jm. An alternative symbol for the direct product is C=AB.
The ordering in rectangular arrays is convenient for dealing with matrices.
The pair ik labels the rows, the pair jm labels the columns of the rectangular array of
lArlBr rows and lAclBc columns of C.
A convenient definition of multiplication of the direct-product matrices results from the requirement of
the representation of “transformations” by successive multiplications of the matrices:
A’’=A’A represents the operation A succeeded by the operation A’;
similarly B’’=B’B, and C’’=C’C=A’AB’B. The elements of the direct product are
' ' ' ' '
, , ,( ' ) ,    ik jm ip pj kq qm ip kq pj qm ik pq pq jm
p q p q p q
C C A A B B A B A B C C
which is the usual “row-times-column” multiplication of the matrices C’ and C.
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The rectangular array of the elements of EB might be visualized as follows:
11 12 1
21 22 2
1 2
...
...
,
.
...
 
 
 
   
 
  
Ac
Ac
Ar Ar Ar Ac
l
l
l l l l
A B A B A B
A B A B A B
A B
A B A B A B
where B is the rectangular block
11 12 1
21 22 2
1 2
...
...
.
.
...
 
 
 
  
 
  
Bc
Bc
Br Br Br Bc
l
l
l l l l
B B B
B B B
B
B B B
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Direct product of groups
Two groups,
GA with the elements Ai , i=1,...,nA , and
GB with the elements Bj , j=1,...,nB , such that AiBj=BjAi for all of their elements,
form the direct product group GAGB consisting of all AiBj.
The four group axioms are evidently fulfilled:
1. AiBjAkBl = (AiAk)(BjBl ),
2. the unit element is EAEB,
3. the inverse element of is Ai
-1Bj
-1 , since Ai
-1Bj
-1AiBj=EAEB,
4. the multiplication is associative.
If GA and GB have no common elements (except possibly for the identity),
the order of GAGB is nAnB.
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Direct product of groups – example
The symmetry operations of an equilateral triangle
(Schoenflies notation)
form the point group C3v {E,3sv ,2C3} if the upper and lower faces are distinguishable;
if this asymmetry is removed, there is another symmetry operation:
sh , the mirror reflection in the horizontal plane.
Since shsh = E, the group C1h {E,sh} is a cyclic group of order 2.
The horizontal reflection sh commutes with any element of C3v, and the complete
symmetry of the equilateral triangle is described by the group
D3h = C3v C1h with the 12 elements
{E, s1, s2, s3, C3, C3
2, sh, shs1, shs2, shs3 , shC3, shC3
2}.
3
y
x
1
2
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Direct product of groups – example of D3h = C3v C1h , the multiplication table
simpler notation of P(3): s1A,s2B,s3C,C3D,C3
2F; further, shS:
right
left
E A B C D F
E E A B C D F
A A E D F B C
B B F E D C A
C C D F E A B
D D C A B F E
F F B C A E D
E S
E E S
S S E
right
left
E A B C D F S SA SB SC SD SF
E E A B C D F ?
A A E D F B C
B B F E D C A ?
C C D F E A B ?
...
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Direct product of groups – example of D3h = C3v C1h , classes, irreducible reps
six classes:
{E}, {s1, s2, s3},{C3,C3
2},
{sh},{shs1, shs2, shs3},{shC3, shC3
2}
we are interested in the 6x6 matrix of characters of irreducible representations
of the direct product
(character tables from Inui, Tanabe, Onodera, Group theory and its applications in physics, Springer 1976)
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Direct product of groups – example of D3h = C3v C1h , characters of irreps
the 3x3 matrix of characters of D3h is 3x repeated, and the lower diagonal block
is of the opposite sign due to the second irreducible representation of C1h , with
the character A’’=(1,-1)
15
16
Direct product of groups – example of D3h = C3v C1h , classes, irreducible reps
a different notation for some of the classes
(character tables from M.S. Dresselhaus, G. Dresselhaus, A. Jorio, Group theory, Applications to the physics of
Condensed Matter, Springer 2008)
the rows and columns are distinct (in fact, orthogonal)  we can find
the correspondence with the previous version of the table (the are identical
as far as the characters are concerned)
Point groups: notation for the representations
Chemical (Mulliken,1933) notation is common in molecular physics or in lattice
dynamics. It uses
A and B for one-dimensional representations (B if odd under the smallest rotation
of the principal axis),
E for two-dimensional representations,
T,U,V,W for the dimensionalities of 3,4,5,6.
Physical (Bethe, 1929; Koster, Dimmock, Wheeler and Statz, 1963) notation:
G1, G2, G3,... ; preferred in the recent solid-state literature.
An alternative (Bouckaert, Smoluchowski and Wigner, 1935) is available on occasion;
example for Td:
Mulliken KDWS BSW
A1 G1 G1
A2 G2 G2
E G3 G12
T1 G4 G15
T2 G5 G25
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Point groups: notation for the representations
The Mulliken notation has an additional rule:
if the group contains inversion, the symbol has an additional suffix, either
“g” (gerade) for the even parity under inversion, or
“u” (ungerade) for the odd parity.
The example of the orthorhombic point group D2h=D2CI , CI ={E,I} is a cyclic
group of order 2.
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Symmetry operations on functions of coordinates
Consider a rotation by the angle a in the (x,y) plane,
1' cos sin cos sin cos sin
( ) , ( ) , ( ) .
' sin cos sin cos sin cos
x x y x
R R R
y x y y
a a a a a a
a a a
a a a a a a
          
                      
This transformation of the coordinates transforms also their functions, f(x,y), such
as f1(x,y)=x, f2(x,y)=x2+ y2, f3(x,y)=x2-y2, f4(x,y)=xy, f5(x,y)=x3-3xy2,...
The transformed function values are given by
'( ', ') ( , ),f x y f x y
and the transformed function results from the original one by the action of an
operator PR (acting on functions):
' , ( ', ') ( , ) ( 'cos 'sin , 'cos 'sin ).R Rf P f P f x y f x y f x y y xa a a a    
The explicit form for the transformed function is therefore
( , ) ( cos sin , cos sin ).RP f x y f x y y xa a a a  
19
Symmetry operations on functions of coordinates
The rotation Ra transforms the complex-valued function fc1(x,y)=x+iy into
With f2(x,y)=x2+ y2, f3(x,y)=x2-y2, f4(x,y)=xy, we obtain the following examples
of the transformations:
1 1( , ) cos sin ( cos sin ) ( , ).i
R c cP f x y x y i y x e f x ya
a a a a 
    
2 2
2 2
2 2 2 2 2 2
4 3 4
' ,
' cos sin ( ) (cos sin ) cos sin (cos sin ) .
f x y f
f x y xy f fa a a a a a a a
  
        
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Symmetry operations on functions of coordinates
For any transformation R of the 3-dimensional vector r=(x,y,z), r’=Rr, we obtain
the transformed function from the generalized recipe:
1
1
( ') ( ) ( '), i.e.,
( ) ( ).
R
R
P f f f R
P f f R


 

r r r
r r
Two successive operations R and S transform an arbitrary function f in the
following way:
1 1 1
( ) [ ( )] ( ) ( ) ( ),S R S R SP P f P P f P g g S f R S  
   r r r r r
where g=PRf.
The combined action of the operation R (applied first) and S is the product SR:
1 1 1
( ) [( ) ] ( ),SRP f f SR f R S  
 r r r
leading to the identical result as the product PSPR. Consequently, it is possible to
use the same symbol for the operations R and PR:
1
( ) ( ).Rf f R
r r
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Basis functions of a representation
The set of independent functions f1, f2, ..., fd is called a basis of a d-dimensional
representation, formed by the matrices with the elements Dkl(Ai), if
1
( ) for any .

 
d
i l kl i k i
k
A f D A f A G
This is the condition of the closure of the set of functions under the operations of
the group G.
Individual functions of this set are called basis functions, partners, or basis vectors.
The l-th partner results from the linear combination with the coefficients from the
l-th column of the set of representation matrices; it belongs to the l-th column.
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A (reducible) 3-dimensional representation of P(3) can be used as the following
transformation of f1=x, f2=y, f3=z by the elements of C3v:
E C3 C3
-1
1 0 0
0 1 0
0 0 1
     
          
          
x x
y y
z z
1 0 0
0 0 1
0 1 0
     
          
          
x x
z y
y z
0 0 1
0 1 0
1 0 0
     
          
          
z x
y y
x z
s1 s2 s3
0 1 0
1 0 0
0 0 1
     
          
          
y x
x y
z z
0 0 1
1 0 0
0 1 0
     
          
          
z x
x y
y z
0 1 0
0 0 1
1 0 0
     
          
          
y x
z y
x z
23
Its character is
P3=A1+E,
it is orthogonal to A2
(the projection on A2
vanishes)
E 3sv 2C3
P3 3 1 0
A1 1 1 1
A2 1 -1 1
E 2 0 -1
The function
1 1 2 3     Af f f f x y z
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remains invariant under all operations of C3v;
if forms a basis for the representation A1, or it transforms as A1.
Similarly, the functions
1 2(2 )/ 6, ( )/ 2    E Ef x y z f y z
form the basis for the irreducible representation E.
A basis for the representation A2 can be obtained from polynomials of the third
order:
2
2 2 2 2 2 2
( ) ( ) ( ).     Af x y z y z y z x y